Multiscale difference equation signal modeling and analysis techniques

A novel signal modeling technique in which an arbitrary sequence of data points is represented as samples of the solution to a multiscale difference equation is proposed. Such a model completely characterizes a number of higher derivatives of the signal as well as the signal itself. It provides a recursive signal interpolation mechanism as a function of scale. It also leads to multigrid type signal filtering, detection and estimation algorithms. The existence and uniqueness of L₁ and L₂ solutions to the multiscale difference equation are first investigated. The paper generalizes the results obtained for the two scale difference equation. Next, we provide conditions for the existence of unique solution to such an equation. Finally, techniques for modeling an arbitrary set of data samples as samples of the solution to a multiscale difference equation are presented. Specifically, we describe the encoding and decoding steps in these techniques. We also prevent audio signal modeling examples to illustrate multiscale difference equation models